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1. A triangulation of a polygon is a way to divide the polygon into triangles by adding diagonals that do not
intersect each other (except possibly at the vertices of the polygon). How many diagonals does one need to add
in order to triangulate a regular 2021-gon?

 

2. Let a, b, c, d, and e be five consecutive terms in an arithmetic sequence, and suppose that a + b + c + d + e = 40.
Which of a, b, c, d, or e can be found? (ITS NOT 6+7+8+9+10, it can only be one of the letters.)

 

3.A chef places four tablespoons of water into an eight-tablespoon measuring cup and four tablespoons of vinegar
into a second cup of the same size. She then pours half the water from the first cup to the second and, after
stirring thoroughly, pours half the liquid in the second cup back to the first. What fraction of the liquid in the first
cup is now vinegar?

 Jan 30, 2021
edited by Guest  Jan 30, 2021
 #1
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1. From the formula, a regular 2021-gon has 2021*2018/2 = 2039189 diagonals.

 

2. You can find a, becuase a is part of the formula for the sum

a + (a + d) + (a + 2d) + ... + (a + (n - 1)d) = (2a + (n - 1)d)/2.

 Jan 30, 2021
 #2
avatar+120023 
+1

2. Let a, b, c, d, and e be five consecutive terms in an arithmetic sequence, and suppose that a + b + c + d + e = 40.
Which of a, b, c, d, or e can be found? 

 

c, the middle term,  is the one  that  can  be  found

 

We have an odd  number of terms..so this will   = the  sum of the terms   /  the number of  terms   =    40 / 5 =     8

 

To see this...suppose  we  have

 

1 +  4  + 7  + 10   +  13  =   35

 

The middle  term is  35/5   = 7

 

 

 

cool cool cool

 Jan 30, 2021
 #3
avatar+120023 
+1

3.A chef places four tablespoons of water into an eight-tablespoon measuring cup and four tablespoons of vinegar
into a second cup of the same size. She then pours half the water from the first cup to the second and, after
stirring thoroughly, pours half the liquid in the second cup back to the first. What fraction of the liquid in the first
cup is now vinegar?

 

 

After pouring,  the  first cup contains   2 Tsp  of water

 

And the mixture in the second  cup   is    2 parts water and  4 parts  vinegar  =  2 Tsp of water and 4 Tsp of vinegar

 

The  mxing does not affect the ratio

 

So..... she  pours  1/2 of this back into the  first  cup    = 1 Tsp of water and 2 Tsp of  vinegar

 

And the first  cup  now contains     3 Tsp of water  and 2 Tsp  of  vinegar

 

So....the first cup   contains      2 / ( 2 + 3)  =    2/5  vinegar

 

 

cool cool cool

 Jan 30, 2021

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